Digital Signal Processing Reference
In-Depth Information
∴
Due to normalization,
P
δ
(
f
)
)
=
p
(0)
=
1.
(5.11)
Now, equating the R.H.S. of Eqs. (
5.8
) and (
5.11
),
∞
1
T
B
P
(
f
−
nR
B
)
=
1
n
=−∞
(5.12)
∞
Z
(
f
)
=
P
(
f
−
nR
B
)
=
T
B
n
=−∞
The expression is called as
Nyquist criterion for distortion less transmission.
5.2.2 Concept of Ideal Nyquist Channel
From the expression of 'Nyquist criterion for distortion less transmission', the
L.H.S. of Eq. (
5.12
) represents a series of shifted spectrums [
7
]. For n
0, the
L.H.S. corresponds to P(f) and it represents a frequency function with narrowest
band which satisfies the aforementioned equation.
Suppose that, the channel has a bandwidth of B Hz. Then,
P(f)
=
=
0, for
|
f
|
>
B
.
We distinguish three cases as follows.
∞
1. In this case,
T
B
<
1
/
2
B
or equivalently, 1
/
T
B
>
2
B
. Since,
P
(
f
−
nR
B
)
n
=−∞
separated by 1/T
B
as in Fig.
5.3
,
there is no choice of P(f) to ensure the condition of Eq. (
5.12
) in this case. Thus,
we cannot design a zero ISI system considering case 1.
2. In this case,
T
B
contains non-overlapping replicas of
P
(
f
)
)
=
1
/
2
B
or equivalently, 1
/
T
B
=
2
B
(Nyquist rate). Here, the
replicas of
P
are about to overlap, as they are separated by 1/T
B
as in
Fig.
5.4
. Therefore, there exists only one
P
(
f
)
)
(
f
)
)
that results in
P
δ
(
f
)
)
=
1, i.e.,
∞
P
(
f
−
nR
B
)
=
T
B
as
n
=−∞
T
B
;
|
|
<
f
B
P
(
f
)
)
=
(5.13)
0;
otherwise
In other words,
f
2
B
2
B
rect
f
,
1
P
(
f
)
)
=
T
B
=
2
B
which results in
sin
c
t
T
B
F
−
1
P
)
=
=
(
=
(
)
p
(
t
)
f
)
sin
c
2
Bt
(5.14)
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